The Fourier transform of a function ƒ(t) has served as the most important transform in numerous signal processing applications. For example, the Fourier transform is widely used in imaging analysis such as CT and Magnetic Resonance Imaging (MRI).
Standard Fourier analysis reveals individual frequency components involved in a signal or image. However, in many situations of frequencies changing over time or space the standard Fourier analysis does not provide sufficient information. In numerous applications processing of non-stationary signals or images reveals important information. For example, in MRI signal processing motion caused by respiratory activity, cardiac activity, blood flow causes temporal changes in spatial frequencies.
To overcome the deficiency of the standard Fourier analysis, other techniques such as the Gabor transform (GT) disclosed in: Gabor, D. “Theory of communications”, J. Inst. Elec. Eng., 1946; 93, 429-457, also known as the short time Fourier transform, and the Wavelet transform (WT) disclosed in: Goupillaud P., Grossmann, A., Morlet J. “Cycle-octave and related transforms in seismic signal analysis”, Geoexplor, 1984; 23, 85-102,and in: Grossmann, A., Morlet J. “Decomposition of Hardy functions into square integrable Wavelets of constant shape”, SIAM J. Math. Anal.,1984; 15, 723-736,have been developed, references to which are incorporated herein by reference. Both of these methods unfold the time information by localizing the signal in time and calculating its “instantaneous frequencies.” However, both the GT and the WT have limitations substantially reducing their usefulness in the analysis of imaging signal data. The GT has a constant resolution over the entire time-frequency domain which limits the detection of relatively small frequency changes. The WT has variant resolutions, but it provides time vs. scale information as opposed to time vs. frequency information. Although “scale” is loosely related to “frequency”—low scale corresponds to high frequency and high scale to low frequency—for most wavelets there is no explicit relationship between scale factors and the Fourier frequencies. Therefore, the time-scale representation of a signal is difficult if not impossible to interpret.
The Stockwell transform (ST) disclosed in: Stockwell R. G., Mansinha L., Lowe R. P., “Localization of the complex spectrum: the S-transform”, IEEE Trans. Signal Process, 1996; 44, 998-1001,references to which are incorporated herein by reference, is a spectral localization transform that utilizes a frequency adapted Gaussian window to achieve optimum resolution at each frequency.
While the standard Fourier transform provides information about the frequency content of an entire signal or image the ST provides a local spectrum for each point of the signal or image. Therefore, the ST provides information about changes in frequency content over time or space. The 1D ST applied to signal data such as time course fMRI data is used to localize and remove noise components and artifacts. The 2D ST provides local textural information for each point in an image. This information is useful in distinguishing between tissues of differing appearance. For example, a texture map of an MR image enhances differences indicating lesions or other abnormalities due to disease activity that are difficult to distinguish in conventional MR images.
However, the ST of a 2D image function I(x, y) retains spectral variables kx and ky as well as spatial variables x and y, resulting in a complex-valued function of four variables. Therefore, calculation of the S transform of multidimensional signal data is computationally intensive. Unfortunately, the processing time needed for transforming a typical multidimensional MR image is preventing the S transform from numerous practical applications in a clinical setting.